Spells Known: This is the number of different spells that you have access to from the spell list for your class. Some classes (like the Wizard) only know spells they have researched in some way, though they may add new spells known as they come across them. Some classes (like the Cleric) automatically know all of the spells in their class spell list. Still other classes (like the Bard) gain knowledge of new spells innately when they gain a level and know a set number of spells per level from their class list. You must know a spell in order to be able to prepare that spell for casting, so this is really step one of casting a spell.

Spells Prepared: Once you know a spell, you must specifically prepare that spell in order to have it available to cast. Some classes (like the Bard) automatically have all of their known spells prepared. Most classes (like the Wizard and Cleric) must choose a number of spells from their known spells to prepare. In all cases, the number of spells that can be prepared is determined by your level in the spellcasting class. In contrast to previous editions, your prepared spells remain prepared until you elect to change them - casting a particular spell does not erase it from your prepared spells list.

Spell Slots: Spell slots are the resource used for casting spells. For example, if my character has four 1st level spell slots and two 2nd level spell slots, then he can cast a total of six spells before resting. Once a spell is cast using a spell slot, then that spell slot is expended and cannot be used again until after a long rest. A spell slot can be used to cast a spell of that level or a spell of a lower level. This is a little different than previous editions, in that the Prepared Spells are essentially the pool of available spells while the Spell Slots represent the ability to cast them. Also, a caster can use a higher level slot to cast a lower level spell which will often increase the effectiveness of that spell.

For example, let us say that we have a level 5 Wizard. The Wizard has the following:

On this particular day, he can choose to cast Magic Missile nine times (and then all of his spell slots would be used up and he could cast nothing else until completing a long rest). He can cast it four times as a 1st level spell, creating three darts each doing 1d4+1 damage. He can then cast it three times from a 2nd level slot and two times from a 3rd level slot. Looking a the description, if cast in a higher level slot it creates one additional missile per level. So when cast using a 2nd level slot it would create four darts (each doing 1d4+1), and when cast from a 3rd level slot it would create five darts. Not all spells gain additional effects for being cast from a higher level slot - for example, Mage Armor can be cast using a 2nd or 3rd level slot but it functions exactly the same as it would in a 1st level slot.

Or, he could cast Mage Armor twice using up two 1st level slots, Magic Missile twice using two 1st level slots, Magic Missile From a 2nd level slot, Mirror Image from a 2nd level slot, Melf's Acid Arrow from a 2nd Level Slot, Melf's Acid Arrow from a 3rd Level Slot, and Animate Dead from a 3rd level slot. Basically, he can cast any combination of spells prepared using the spell slots he has available. The only caveat is that a higher level spell cannot be cast in a lower level slot (ie. Fly, a 3rd level spell, cannot be cast in a slot lower than 3rd level).

All in all, it seems a bit more flexible than it has been in the past.

Generally speaking, ability checks are used when attempting any action (other than an attack) that has a chance of failure. The GM determines which ability is used for the check and if a particular skill applies or not. To make an ability check, roll a d20 and add the appropriate ability modifier. If a skill applies and you are proficient in that skill then you also add your proficiency bonus to the check.

There is also a concept of passive checks - ability checks that do not involve die rolls on the part of the player. They are typically used as an average for something done repeatedly (searching over and over for a secret door) or when the GM wants to resolve a roll secretly (such as whether or not you spot a hidden monster). A character's total for the passive check is 10 + ability modifier + proficiency bonus (if you have proficiency in the appropriate skill).

Typical ability checks:
Strength - Applying brute force to a situation (breaking, bending, lifting, pushing, gripping)
Dexterity - Moving nimbly/quickly/quietly, avoiding falling, dodging out of the way of something, wriggling out of bonds or tight spaces
Constitution - Trying to push yourself past your limits, such as holding your breath, forced marching, or going without sleep
Intelligence - Using logic, education, memory, or deduction to solve a problem including such things as forgery, creating a disguise, appraising an item
Wisdom - How attuned you are to the world, get a gut feeling about a course of action or situation
Charisma - Ability to interact effectively with others, know who to talk to for news rumors and gossip, get an idea of important topics among people